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Quaternions, Octonions, and the Quantization of Games

Aden Ahmed

Published Apr 20, 2009• 1 citation
Publisher

Abstract

We present an effect on classical games that is obtained by replacing the notion of probability distribution with the notions of quantum superposition and measurement. Our particular focus will be on two and three player games where each player has precisely two pure strategic choices. Games in normal form are represented as "payoff" functions. Game quantization requires the extension of these functions to much larger domains. The main result of this work is the co-ordinatization of these extended functions by either the quaternions or octonions in order to obtain computationally friendly versions of these functions. This computational capability is then exploited to analyze and potentially classify the Nash equilibria in the new extended games with occasionally counter intuitive results.

DOI
10.15760/etd.7814

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We present an effect on classical games that is obtained by replacing the notion of probability distribution with the notions of quantum superposition and measurement. Our particular focus will be on two and three player games where each player has precisely two pure strategic choices. Games in normal form are represented as "payoff" functions. Game quantization requires the extension of these functions to much larger domains. The main result of this work is the co-ordinatization of these extended functions by either the quaternions or octonions in order to obtain computationally friendly versions of these functions. This computational capability is then exploited to analyze and potentially classify the Nash equilibria in the new extended games with occasionally counter intuitive results.
After a brief exploration of certain topological features of the unit octonions, we identify the Nash equilibria in a quantized version of three-player dilemma game using an octonionic representation of the payoff function. This representation is an important tool for the relative ease in working with octonionic arithmetic as opposed to multi-variant tensors and provides a fresh computational framework that can be utilized for the classification of Nash equilibria in generic three-player two-strategy games. While the full classification of equilibria remains a goal of future research, our representation has already established certain family of equilibria in a quantized version of three-player dilemma game.
A game is said to be “quantized" when the expected payoff to the player(s) is computed via the higher order randomization notion of quantum superposition followed by measurement versus the randomization notion of probability distribution. A major motivation for quantizing a game is the potential manifestation of Nash equilibria that are superior to those already available in the game. Quantum superpositions are elements of a (projective) Hilbert space which, among other things, is an inner product space. The inner product of the Hilbert space of quantum superpositions is used here to give a geometric characterization of Nash equilibrium in quantized versions of Hawk-Dove games, a class of games to which the well known game Prisoners
The concept of Nash equilibria in pure strategies for quantum extensions of the general form of the Prisoner's Dilemma game is investigated. The process of quantization involves incorporating two additional unitary strategies, which effectively expand the classical game. We consider five classes of such quantum games, which remain invariant under isomorphic transformations of the classical game. The resulting Nash equilibria are found to be more closely aligned with Pareto-optimal solutions than those of the conventional Nash equilibrium outcome of the classical game. Our results demonstrate the complexity and diversity of strategic behavior in the quantum setting, providing new insights into the dynamics of classical decision-making dilemmas. In particular, we provide a detailed characterization of strategy profiles and their corresponding Nash equilibria, thereby extending the understanding of quantum strategies' impact on traditional game-theoretical problems.
For any two-by-two game $\G$, we define a new two-player game $\G^Q$. The definition is motivated by a vision of players in game $\G$ communicating via quantum technology according to a certain standard protocol originally introduced by Eisert and Wilkins [EW]. In the game $\G^Q$, each players' strategy set consists of the set of all probability distributions on the 3-sphere S3S^3S3. Nash equilibria in this game can be difficult to compute. Our main theorems classify all possible equilibria in $\G^Q$ for a Zariski-dense set of games $\G$ that we call {\it generic games}. First, we show that up to a suitable definition of equivalence, any strategy that arises in equilibrium is supported on at most four points; then we show that those four points must lie in one of a small number of geometric configurations. One easy consequence is that for zero-sum games, the payoff to either player in a mixed strategy quantum equilibrium must equal the average of that player's four possible payoffs.
This paper investigates Nash equilibria in pure strategies for quantum approach to the Prisoner's Dilemma. The quantization process involves extending the classical game by introducing two additional unitary strategies. We consider five classes of such quantum games, which remain invariant under isomorphic transformations of the classical game. For each class, we identify and analyse all possible Nash equilibria. Our results reveal the complexity and diversity of strategic behaviour in the quantum setting, providing new insights into the dynamics of classical decision-making dilemmas. In the case of the standard Prisoner's Dilemma, the resulting Nash equilibria of quantum extensions are found to be closer to Pareto optimal solutions than those of the classical equilibrium.